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Theorem archiabl 29537
Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
Assertion
Ref Expression
archiabl ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Proof of Theorem archiabl
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . . 5 (0g𝑊) = (0g𝑊)
3 eqid 2621 . . . . 5 (le‘𝑊) = (le‘𝑊)
4 eqid 2621 . . . . 5 (lt‘𝑊) = (lt‘𝑊)
5 eqid 2621 . . . . 5 (.g𝑊) = (.g𝑊)
6 simpll1 1098 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
7 simpll3 1100 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
8 simplr 791 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑣 ∈ (Base‘𝑊))
9 simprl 793 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → (0g𝑊)(lt‘𝑊)𝑣)
10 simp2 1060 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑦 ∈ (Base‘𝑊))
11 simp1rr 1125 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
12 simp3 1061 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → (0g𝑊)(lt‘𝑊)𝑦)
13 breq2 4617 . . . . . . . 8 (𝑥 = 𝑦 → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑦))
14 breq2 4617 . . . . . . . 8 (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥𝑣(le‘𝑊)𝑦))
1513, 14imbi12d 334 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1615rspcv 3291 . . . . . 6 (𝑦 ∈ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) → ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1710, 11, 12, 16syl3c 66 . . . . 5 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑣(le‘𝑊)𝑦)
181, 2, 3, 4, 5, 6, 7, 8, 9, 17archiabllem1 29532 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
1918adantllr 754 . . 3 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
20 simpr 477 . . . 4 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
21 breq2 4617 . . . . . 6 (𝑢 = 𝑣 → ((0g𝑊)(lt‘𝑊)𝑢 ↔ (0g𝑊)(lt‘𝑊)𝑣))
22 breq1 4616 . . . . . . . 8 (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
2322imbi2d 330 . . . . . . 7 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2423ralbidv 2980 . . . . . 6 (𝑢 = 𝑣 → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2521, 24anbi12d 746 . . . . 5 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))))
2625cbvrexv 3160 . . . 4 (∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2720, 26sylib 208 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2819, 27r19.29a 3071 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
29 simpl1 1062 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
30 simpl3 1064 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
31 eqid 2621 . . 3 (+g𝑊) = (+g𝑊)
32 simpl2 1063 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → (oppg𝑊) ∈ oGrp)
33 simpr 477 . . . . . . . . . 10 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
34 ralnex 2986 . . . . . . . . . 10 (∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3533, 34sylibr 224 . . . . . . . . 9 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
36 rexanali 2992 . . . . . . . . . . . 12 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))
3736imbi2i 326 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
38 imnan 438 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3937, 38bitri 264 . . . . . . . . . 10 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4039ralbii 2974 . . . . . . . . 9 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4135, 40sylibr 224 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)))
4222notbid 308 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
4342anbi2d 739 . . . . . . . . . . 11 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4443rexbidv 3045 . . . . . . . . . 10 (𝑢 = 𝑣 → (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4521, 44imbi12d 334 . . . . . . . . 9 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥))))
4645cbvralv 3159 . . . . . . . 8 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4741, 46sylib 208 . . . . . . 7 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4847r19.21bi 2927 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4914notbid 308 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
5013, 49anbi12d 746 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
5150cbvrexv 3160 . . . . . 6 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
5248, 51syl6ib 241 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
53523impia 1258 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
54 simp1l1 1152 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ oGrp)
55 isogrp 29487 . . . . . . 7 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
5655simprbi 480 . . . . . 6 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57 omndtos 29490 . . . . . 6 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
5854, 56, 573syl 18 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ Toset)
59 simp2 1060 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑣 ∈ (Base‘𝑊))
601, 3, 4tltnle 29447 . . . . . . . . . 10 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
6160bicomd 213 . . . . . . . . 9 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
62613com23 1268 . . . . . . . 8 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
63623expa 1262 . . . . . . 7 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
6463anbi2d 739 . . . . . 6 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6564rexbidva 3042 . . . . 5 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6658, 59, 65syl2anc 692 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6753, 66mpbid 222 . . 3 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
681, 2, 3, 4, 5, 29, 30, 31, 32, 67archiabllem2 29536 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
6928, 68pm2.61dan 831 1 ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  cfv 5847  Basecbs 15781  +gcplusg 15862  lecple 15869  0gc0g 16021  ltcplt 16862  Tosetctos 16954  Grpcgrp 17343  .gcmg 17461  oppgcoppg 17696  Abelcabl 18115  oMndcomnd 29482  oGrpcogrp 29483  Archicarchi 29516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-seq 12742  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-ple 15882  df-0g 16023  df-preset 16849  df-poset 16867  df-plt 16879  df-toset 16955  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-oppg 17697  df-cmn 18116  df-abl 18117  df-omnd 29484  df-ogrp 29485  df-inftm 29517  df-archi 29518
This theorem is referenced by: (None)
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